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One-Way BG ANOVA Andrew Ainsworth Psy 420

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Topics Analysis with more than 2 levels Deviation, Computation, Regression, Unequal Samples Specific Comparisons Trend Analysis, Planned comparisons, Post- Hoc Adjustments Effect Size Measures Eta Squared, Omega Squared, Cohen’s d Power and Sample Size Estimates

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Deviation Approach When the n’s are not equal

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Analysis - Traditional The traditional analysis is the same

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Analysis - Traditional Traditional Analysis – Unequal Samples

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Unequal N and DFs

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Analysis - Regression In order to perform a complete analysis of variance through regression you need to cover all of the between groups variance To do this you need to: Create k – 1 dichotomous predictors (Xs) Make sure the predictors don’t overlap

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Analysis – Regression

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One of the easiest ways to ensure that the comps do not overlap is to make sure they are orthogonal Orthogonal (independence) The sum of each comparison equals zero The sum of each cross-product of predictors equals zero

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Analysis – Regression

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Analysis - Regression

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Analysis – Regression

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Formulas

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Analysis – Regression Formulas

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Analysis – Regression Example

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Analysis - Regression Example

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Analysis - Regression Example

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Analysis - Regression Example

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Analysis - Regression Example F crit (2,12) = 3.88, since 12.253 is greater than 3.88 you reject the null hypothesis. There is evidence that drug type can predict level of anxiety

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Analysis - Regression Example

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Analysis - Regression SPSS

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Analysis - Regression SPSS

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Analysis - Regression SPSS

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Specific Comparisons F-test for Comparisons n = number of subjects in each group = squared sum of the weighted means = sum of the squared coefficients MS S/A = mean square error from overall ANOVA

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Specific Comparisons If each group has a different sample size…

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Specific Comparisons Example

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Specific Comparisons Trend Analysis If you have ordered groups (e.g. they differ in amount of Milligrams given; 5, 10, 15, 20) You often will want to know whether there is a consistent trend across the ordered groups (e.g. linear trend) Trend analysis comes in handy too because there are orthogonal weights already worked out depending on the number of groups (pg. 703)

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Specific Comparisons Different types of trend and coefficients for 4 groups

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Specific Comparisons Mixtures of Linear and Quadratic Trend

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Specific Comparisons Planned comparisons - if the comparisons are planned than you test them without any correction Each F-test for the comparison is treated like any other F-test You look up an F-critical value in a table with df comp and df error.

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Specific Comparisons Example – if the comparisons are planned than you test them without any correction… F x1, since 11.8 is larger than 4.75 there is evidence that the subjects in the control group had higher anxiety than the treatment groups F x2, since 12.75 is larger than 4.75 there is evidence that subjects in the Scruital group reporter lower anxiety than the Ativan group

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Specific Comparisons Post hoc adjustments Scheffé This is used for complex comparisons, and is conservative Calculate F comp as usual F S = (a – 1)F C where F S is the new critical value a – 1 is the number of groups minus 1 F C is the original critical value

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Specific Comparisons Post hoc adjustments Scheffé – Example F X1 = 11.8 F S = (3 – 1) * 4.75 = 9.5 Even with a post hoc adjustment the difference between the control group and the two treatment groups is still significant

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Specific Comparisons Post hoc adjustments Tukey’s Honestly Significant Difference (HSD) or Studentized Range Statistic For all pairwise tests, no pooled or averaged means F comp is the same, q T is a tabled value on pgs. 699-700

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Specific Comparisons Post hoc adjustments Tukey’s Honestly Significant Difference (HSD) or Studentized Range Statistic Or if you have many pairs to test you can calculate a significant mean difference based on the HSD, where q T is the same as before, when unequal samples

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Specific Comparisons Post hoc adjustments Tukey’s – example Since 12.74 is greater than 7.11, the differences between the two treatment groups is still significant after the post hoc adjustment

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Specific Comparisons Post hoc adjustments Tukey’s – example Or you calculate: This means that any mean difference above 1.79 is significant according to the HSD adjustment 7.2 – 4.8 = 2.4, since 2.4 is larger than 1.79…

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Effect Size A significant effect depends: Size of the mean differences (effect) Size of the error variance Degrees of freedom Practical Significance Is the effect useful? Meaningful? Does the effect have any real utility?

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Effect Size Raw Effect size – Just looking at the raw difference between the groups Can be illustrated as the largest group difference or smallest (depending) Can’t be compared across samples or experiments

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Effect Size Standardized Effect Size Expresses raw mean differences in standard deviation units Usually referred to as Cohen’s d

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Effect Size Standardized Effect Size Cohen established effect size categories.2 = small effect.5 = moderate effect.8 = large effect

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Effect Size Percent of Overlap There are many effect size measures that indicate the amount of total variance that is accounted for by the effect

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Effect Size Percent of Overlap Eta Squared simply a descriptive statistic Often overestimates the degree of overlap in the population

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Effect Size Omega Squared This is a better estimate of the percent of overlap in the population Corrects for the size of error and the number of groups

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Effect Size Example

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Effect Size For comparisons You can think of this in two different ways SS comp = the numerator of the F comp

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Effect Size For comparisons - Example

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Power and Sample Size Designing powerful studies Select levels of the IV that are very different (increase the effect size) Use a more liberal α level Reduce error variability Compute the sample size necessary for adequate power

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Power and Sample Size Estimating Sample size There are many computer programs that can compute sample size for you (PC-Size, G-power, etc.) You can also calculate it by hand: Where 2 = estimated MS S/A = desired difference Z α-1 = Z value associated with 1 - α z -1 = Z value associated with 1 -

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Power and Sample Size Estimating Sample size – example For overall ANOVA with alpha =.05 and power =.80 (values in table on page 113) Use the largest mean difference Roughly 2 subjects per group For all differences significant Roughly 28 subjects per group

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